Found problems: 85335
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
[b]a)[/b] Let be two nonegative integers $ n\ge 1,k, $ and $ n $ real numbers $ a,b,\ldots ,c. $ Prove that
$$ (1/a+1/b+\cdots 1/c)\left( a^{1+k} +b^{1+k}+\cdots c^{1+k} \right)\ge n\left(a^k+b^k+\cdots +c^k\right) . $$
[b]b)[/b] If $ 1\le d\le e\le f\le g\le h\le i\le 1000 $ are six real numbers, determine the minimum value the expression
$$ d/e+f/g+h/i $$
can take.
Let $ABCD$ and $AB’C’D’$ be equally oriented squares. Prove that the lines $BB_1,CC_1,DD_1$ are concurrent.
If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$.
Prove that there are infinitely many positive integers $a$ such that \[a!+(a+2)!\mid (a+2\left\lfloor\sqrt{a}\right\rfloor)!.\]
[i]Proposed by Navid and the4seasons.[/i]
A circle $ C$ with center $ O.$ and a line $ L$ which does not touch circle $ C.$ $ OQ$ is perpendicular to $ L,$ $ Q$ is on $ L.$ $ P$ is on $ L,$ draw two tangents $ L_1, L_2$ to circle $ C.$ $ QA, QB$ are perpendicular to $ L_1, L_2$ respectively. ($ A$ on $ L_1,$ $ B$ on $ L_2$). Prove that, line $ AB$ intersect $ QO$ at a fixed point.
[i]Original formulation:[/i]
A line $ l$ does not meet a circle $ \omega$ with center $ O.$ $ E$ is the point on $ l$ such that $ OE$ is perpendicular to $ l.$ $ M$ is any point on $ l$ other than $ E.$ The tangents from $ M$ to $ \omega$ touch it at $ A$ and $ B.$ $ C$ is the point on $ MA$ such that $ EC$ is perpendicular to $ MA.$ $ D$ is the point on $ MB$ such that $ ED$ is perpendicular to $ MB.$ The line $ CD$ cuts $ OE$ at $ F.$ Prove that the location of $ F$ is independent of that of $ M.$
Find all integers $N$ with the following property: for $10$ but not $11$ consecutive positive integers, each one is a divisor of $N$.
Let $f(x)=x^2-kx+(k-1)^2$ for some constant $k$. What is the largest possible real value of $k$ such that $f$ has at least one real root?
[i]2020 CCA Math Bonanza Individual Round #5[/i]
Which formula represents n-butane?
$ \textbf{(A) }\ce{CH3CH2CH2CH3} \qquad\textbf{(B) } \ce{CH2=CHCH2CH3} \qquad\textbf{(C) } \ce{(CH3)2CHCH3} \qquad\textbf{(D) } \ce{(CH3)3CH}\qquad $
A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.
Let $\{A_n | n = 1, 2, \cdots \} $ be a set of points in the plane such that for each $n$, the disk with center $A_n$ and radius $2^n$ contains no other point $A_j$ . For any given positive real numbers $a < b$ and $R$, show that there is a subset $G$ of the plane satisfying:
[b](i)[/b] the area of $G$ is greater than or equal to $R$;
[b](ii) [/b]for each point $P$ in $G$, $a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.$
The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.
Let $R$ be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of $R$ ?
$\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}7 \qquad \textbf{(E)}\hspace{.05in}8 $
[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway.
[b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus
[b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$.
[b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay?
[b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C.
[b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].
Given an acute triangle $ABC$ with circumcenter $O$. The circumcircle of $BCH$ and a circle with diameter of $AC$ intersect at $P (P \neq C)$. A point $Q$ on segment of $PC$ such that $PB = PQ$. Prove that $\angle ABC = \angle AQP$
In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle.
[i]Proposed by L. Emelyanov[/i]
On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove
$ AB\plus{}BC\plus{}CA\geq 8DE$
We call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.png[/img]
According to the estimated number of participants who gave a correct solution, this was the hardest (!) problem from today's paper. So here is this great German killer - be warned!
Given a circle k and three pairwisely distinct points A, B, C on this circle. Let h and g be the perpendiculars to the line BC at the points B and C. The perpendicular bisector of the segment AB meets the line h at a point F; the perpendicular bisector of the segment AC meets the line g at a point G.
Prove that the product $BF\cdot CG$ is independent from the position of the point A, as long as the points B and C stay fixed.
The actual problem behind the problem: Why on hell should the points B and C stay fixed?
Darij
Compute the sum of all the roots of $ (2x \plus{} 3)(x \minus{} 4) \plus{} (2x \plus{} 3)(x \minus{} 6) \equal{} 0$.
$ \textbf{(A)}\ 7/2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 13$
For all $n>1$, let $f(n)$ be the biggest divisor of $n$ except itself. Does there exists a positive integer $k$ such that the equality $n-f(n)=k$ has exactly $2023$ solutions?
Let $ABC$ be an acute, scalene triangle with orthocentre $H$. Let $\ell_a$ be the line through the reflection of $B$ with respect to $CH$ and the reflection of $C$ with respect to $BH$. Lines $\ell_b$ and $\ell_c$ are defined similarly. Suppose lines $\ell_a$, $\ell_b$, and $\ell_c$ determine a triangle $\mathcal T$.
Prove that the orthocentre of $\mathcal T$, the circumcentre of $\mathcal T$, and $H$ are collinear.
[i]Fedir Yudin, Ukraine[/i]
\(ABCD\) is a inscribed quadrilateral.
\(P\) is the intersection point of its diagonals.
\(O\) is its circumcenter.
\(\Gamma\) is the circumcircle of \(ABO\).
\(\Delta\) is the circumcircle of \(CDO\).
\(M\) is the midpoint of arc \(AB\) on \(\Gamma\) who doesn't contain \(O\).
\(N\) is the midpoint of arc \(CD\) on \(\Delta\) who doesn't contain \(O\).
Show that \(M,N,P\) are collinear.
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$
$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 3a^2 \qquad\textbf{(C)}\ 27a^2 \qquad\textbf{(D)}\ 2a^{3b} \qquad\textbf{(E)}\ 3a^{2b} $
Two equal parallel chords are drawn $ 8$ inches apart in a circle of radius $ 8$ inches. The area of that part of the circle that lies between the chords is:
$ \textbf{(A)}\ 21\frac {1}{3}\pi \minus{} 32\sqrt {3} \qquad\textbf{(B)}\ 32\sqrt {3} \plus{} 21\frac {1}{3}\pi \qquad\textbf{(C)}\ 32\sqrt {3} \plus{} 42\frac {2}{3}\pi$
$ \textbf{(D)}\ 16\sqrt {3} \plus{} 42\frac {2}{3}\pi \qquad\textbf{(E)}\ 42\frac {2}{3}\pi$